Exercise 14

This exercise presents another occasion where Green's Theorem can be used to convert a double integral into a line integral.

1. Recall from Applications of Double Integrals that the centroid of a lamina $D$ of area $A$ is given by $$\begin{gathered} \overline{x} = \frac{1}{A}\iint_D x\, dA \qquad \text{and}\qquad \overline{y} = \frac{1}{A}\iint_D y\, dA \end{gathered}$$. Find vector fields $\vF$ and $\vG$ so that $$\begin{gathered} \overline{x} = \oint_{C} \vF\cdot d\vr \qquad \text{and}\qquad \overline{y} = \oint_C \vG\cdot d\vr \end{gathered}$$, where $C$ is the boundary of the lamina $D$.

2. Find the centroid of the triangle with vertices $(0,0)$, $(a,0)$, and $(0,b)$ for real numbers $a,b>0$.